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Data-Enabled Predictive Control
of Autonomous Energy Systems
Florian D¨
orfler
ETH Z¨
urich
KIOS 2024
Graduate
School
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Acknowledgements
Jeremy Coulson
John Lygeros
Linbin Huang
Ivan Markovsky
Further:
Ezzat Elokda,
Paul Beuchat,
Daniele Alpago,
Jianzhe (Trevor) Zhen,
Claudio de Persis,
Pietro Tesi,
Henk van Waarde,
Eduardo Prieto,
Saverio Bolognani,
Andrea Favato,
Paolo Carlet,
Andrea Martin,
Luca Furieri,
Giancarlo Ferrari-Trecate,
Keith Mo at,
...
& many master students
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Thoughts on data in control systems
increasing role of data-centric methods
in science / engineering / industry due to
• methodological advances in statistics,
optimization, & machine learning (ML)
• unprecedented availability of brute force:
deluge of data & computational power
• ...and frenzy surrounding big data & ML
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Thoughts on data in control systems
increasing role of data-centric methods
in science / engineering / industry due to
• methodological advances in statistics,
optimization, & machine learning (ML)
• unprecedented availability of brute force:
deluge of data & computational power
• ...and frenzy surrounding big data & ML
Make up your own opinion, but ML works
too well to be ignored
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Thoughts on data in control systems
increasing role of data-centric methods
in science / engineering / industry due to
• methodological advances in statistics,
optimization, & machine learning (ML)
• unprecedented availability of brute force:
deluge of data & computational power
• ...and frenzy surrounding big data & ML
Make up your own opinion, but ML works
too well to be ignored – also in control ?!?
3/53
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Thoughts on data in control systems
increasing role of data-centric methods
in science / engineering / industry due to
• methodological advances in statistics,
optimization, & machine learning (ML)
• unprecedented availability of brute force:
deluge of data & computational power
• ...and frenzy surrounding big data & ML
Make up your own opinion, but ML works
too well to be ignored – also in control ?!?
“ One of the major developments in control
over the past decade – & one of the most
important moving forward – is the interaction
of ML & control systems. ” [CSS roadmap]
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Slide 7 text
Thoughts on data in control systems
increasing role of data-centric methods
in science / engineering / industry due to
• methodological advances in statistics,
optimization, & machine learning (ML)
• unprecedented availability of brute force:
deluge of data & computational power
• ...and frenzy surrounding big data & ML
Make up your own opinion, but ML works
too well to be ignored – also in control ?!?
“ One of the major developments in control
over the past decade – & one of the most
important moving forward – is the interaction
of ML & control systems. ” [CSS roadmap]
3/53
Slide 8
Slide 8 text
Thoughts on data in control systems
increasing role of data-centric methods
in science / engineering / industry due to
• methodological advances in statistics,
optimization, & machine learning (ML)
• unprecedented availability of brute force:
deluge of data & computational power
• ...and frenzy surrounding big data & ML
Make up your own opinion, but ML works
too well to be ignored – also in control ?!?
“ One of the major developments in control
over the past decade – & one of the most
important moving forward – is the interaction
of ML & control systems. ” [CSS roadmap]
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Approaches to data-driven control
• indirect data-driven control via models:
data SysID
! model + uncertainty ! control
data-driven
control
u2
u1 y1
y2
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Approaches to data-driven control
• indirect data-driven control via models:
data SysID
! model + uncertainty ! control
• growing trend: direct data-driven control
by-passing models ...(again) hyped, why ?
data-driven
control
u2
u1 y1
y2
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Approaches to data-driven control
• indirect data-driven control via models:
data SysID
! model + uncertainty ! control
• growing trend: direct data-driven control
by-passing models ...(again) hyped, why ?
data-driven
control
u2
u1 y1
y2
Central promise: It is often
easier to learn a control policy
from data rather than a model.
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Approaches to data-driven control
• indirect data-driven control via models:
data SysID
! model + uncertainty ! control
• growing trend: direct data-driven control
by-passing models ...(again) hyped, why ?
data-driven
control
u2
u1 y1
y2
Central promise: It is often
easier to learn a control policy
from data rather than a model.
Example 1973: autotuned PID
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Approaches to data-driven control
• indirect data-driven control via models:
data SysID
! model + uncertainty ! control
• growing trend: direct data-driven control
by-passing models ...(again) hyped, why ?
The direct approach is viable alternative
• for some applications : model-based
approach is too complex to be useful
! too complex models, environments, sensing
modalities, specifications (e.g., wind farm)
data-driven
control
u2
u1 y1
y2
Central promise: It is often
easier to learn a control policy
from data rather than a model.
Example 1973: autotuned PID
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Slide 14 text
Approaches to data-driven control
• indirect data-driven control via models:
data SysID
! model + uncertainty ! control
• growing trend: direct data-driven control
by-passing models ...(again) hyped, why ?
The direct approach is viable alternative
• for some applications : model-based
approach is too complex to be useful
! too complex models, environments, sensing
modalities, specifications (e.g., wind farm)
• due to (well-known) shortcomings of ID
! too cumbersome, models not identified for
control, incompatible uncertainty estimates, ...
data-driven
control
u2
u1 y1
y2
Central promise: It is often
easier to learn a control policy
from data rather than a model.
Example 1973: autotuned PID
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Slide 15 text
Approaches to data-driven control
• indirect data-driven control via models:
data SysID
! model + uncertainty ! control
• growing trend: direct data-driven control
by-passing models ...(again) hyped, why ?
The direct approach is viable alternative
• for some applications : model-based
approach is too complex to be useful
! too complex models, environments, sensing
modalities, specifications (e.g., wind farm)
• due to (well-known) shortcomings of ID
! too cumbersome, models not identified for
control, incompatible uncertainty estimates, ...
• when brute force data/compute available
data-driven
control
u2
u1 y1
y2
Central promise: It is often
easier to learn a control policy
from data rather than a model.
Example 1973: autotuned PID
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Abstraction reveals pros & cons
indirect (model-based) data-driven control
minimize control cost u, x
subject to u, x satisfy state-space model
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↓ inpet
- state
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Abstraction reveals pros & cons
indirect (model-based) data-driven control
minimize control cost u, x
subject to u, x satisfy state-space model
where x estimated from u, y & model
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I / Outpot O
00
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Abstraction reveals pros & cons
indirect (model-based) data-driven control
minimize control cost u, x
subject to u, x satisfy state-space model
where x estimated from u, y & model
where model identified from ud, yd data
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Abstraction reveals pros & cons
indirect (model-based) data-driven control
minimize control cost u, x
subject to u, x satisfy state-space model
where x estimated from u, y & model
where model identified from ud, yd data
! nested multi-level optimization problem
)
outer
optimization
o
middle opt.
o
inner opt.
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Abstraction reveals pros & cons
indirect (model-based) data-driven control
minimize control cost u, x
subject to u, x satisfy state-space model
where x estimated from u, y & model
where model identified from ud, yd data
! nested multi-level optimization problem
)
outer
optimization
o
middle opt.
o
inner opt.
9
>
>
=
>
>
;
separation &
certainty
equivalence
(! LQG case)
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Slide 21 text
Abstraction reveals pros & cons
indirect (model-based) data-driven control
minimize control cost u, x
subject to u, x satisfy state-space model
where x estimated from u, y & model
where model identified from ud, yd data
! nested multi-level optimization problem
)
outer
optimization
o
middle opt.
o
inner opt.
9
>
>
=
>
>
;
separation &
certainty
equivalence
(! LQG case)
no separation
(! ID-4-control)
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Abstraction reveals pros & cons
indirect (model-based) data-driven control
minimize control cost u, x
subject to u, x satisfy state-space model
where x estimated from u, y & model
where model identified from ud, yd data
! nested multi-level optimization problem
)
outer
optimization
o
middle opt.
o
inner opt.
9
>
>
=
>
>
;
separation &
certainty
equivalence
(! LQG case)
no separation
(! ID-4-control)
direct (black-box) data-driven control
minimize control cost u, y
subject to u, y consistent with ud, yd data
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Slide 23
Slide 23 text
Abstraction reveals pros & cons
indirect (model-based) data-driven control
minimize control cost u, x
subject to u, x satisfy state-space model
where x estimated from u, y & model
where model identified from ud, yd data
! nested multi-level optimization problem
)
outer
optimization
o
middle opt.
o
inner opt.
9
>
>
=
>
>
;
separation &
certainty
equivalence
(! LQG case)
no separation
(! ID-4-control)
direct (black-box) data-driven control
minimize control cost u, y
subject to u, y consistent with ud, yd data
! trade-o s
modular vs. end-2-end
suboptimal (?) vs. optimal
convex vs. non-convex (?)
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Slide 24
Slide 24 text
Abstraction reveals pros & cons
indirect (model-based) data-driven control
minimize control cost u, x
subject to u, x satisfy state-space model
where x estimated from u, y & model
where model identified from ud, yd data
! nested multi-level optimization problem
)
outer
optimization
o
middle opt.
o
inner opt.
9
>
>
=
>
>
;
separation &
certainty
equivalence
(! LQG case)
no separation
(! ID-4-control)
direct (black-box) data-driven control
minimize control cost u, y
subject to u, y consistent with ud, yd data
! trade-o s
modular vs. end-2-end
suboptimal (?) vs. optimal
convex vs. non-convex (?)
Additionally: account for uncertainty (hard to propagate in indirect approach)
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Indirect (models) vs. direct (data)
?
x+ = f(x, u)
y = h(x, u)
y
u
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Indirect (models) vs. direct (data)
• models are useful for
design & beyond
• modular ! easy
to debug & interpret
• id = noise filtering
• id = projection on
model class
• harder to propagate
uncertainty through id
• no robust separation
principle ! suboptimal
• ...
?
x+ = f(x, u)
y = h(x, u)
y
u
• some models too
complex to be useful
• end-to-end ! suit-
able for non-experts
• design handles noise
• harder to inject side
info but no bias error
• transparent: no
unmodeled dynamics
• possibly optimal but
often less tractable
• ...
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Slide 27 text
Indirect (models) vs. direct (data)
• models are useful for
design & beyond
• modular ! easy
to debug & interpret
• id = noise filtering
• id = projection on
model class
• harder to propagate
uncertainty through id
• no robust separation
principle ! suboptimal
• ...
?
x+ = f(x, u)
y = h(x, u)
y
u
• some models too
complex to be useful
• end-to-end ! suit-
able for non-experts
• design handles noise
• harder to inject side
info but no bias error
• transparent: no
unmodeled dynamics
• possibly optimal but
often less tractable
• ...
lots of pros, cons, counterexamples, & no universal conclusions [discussion]
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A direct approach: dictionary + MPC
1 trajectory dictionary learning
• motion primitives / basis functions
• theory: Koopman & Liouville
practice: (E)DMD & particles
2 MPC optimizing over dictionary span
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A direct approach: dictionary + MPC
1 trajectory dictionary learning
• motion primitives / basis functions
• theory: Koopman & Liouville
practice: (E)DMD & particles
2 MPC optimizing over dictionary span
! huge theory vs. practice gap
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A direct approach: dictionary + MPC
1 trajectory dictionary learning
• motion primitives / basis functions
• theory: Koopman & Liouville
practice: (E)DMD & particles
2 MPC optimizing over dictionary span
! huge theory vs. practice gap
! back to basics: impulse response
y4
y2
y1 y3 y5
y6
y7
u1 = u2 = · · · = 0
u0 = 1
x0 =0
y0
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A direct approach: dictionary + MPC
1 trajectory dictionary learning
• motion primitives / basis functions
• theory: Koopman & Liouville
practice: (E)DMD & particles
2 MPC optimizing over dictionary span
! huge theory vs. practice gap
! back to basics: impulse response
y4
y2
y1 y3 y5
y6
y7
u1 = u2 = · · · = 0
u0 = 1
x0 =0
y0
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impulse Linear Time-
invariant (ITI)
impulse
Response
u= So 3 y( = g(t =
Ego ,
gg. .
↑
impulse response
↑ spouse to
any other input uH is y(t)
= g(+ -
4) -
u(i)
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y4
y2
y1 y3 y5
y6
y7
u1 = u2 = · · · = 0
u0 = 1
x0 =0
y0
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y4
y2
y1 y3 y5
y6
y7
u1 = u2 = · · · = 0
u0 = 1
x0 =0
y0
Now what if we had the impulse response recorded in our data-library?
⇥
g0 g1 g2 . . .
⇤
=
⇥
yd
0
yd
1
yd
2
. . .
⇤
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response to
any new input Ufuture It is
+-
A
Yfuture (t) =
yd(t-r) . Ufuture (H)
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y4
y2
y1 y3 y5
y6
y7
u1 = u2 = · · · = 0
u0 = 1
x0 =0
y0
Now what if we had the impulse response recorded in our data-library?
⇥
g0 g1 g2 . . .
⇤
=
⇥
yd
0
yd
1
yd
2
. . .
⇤
yfuture(t) =
⇥
yd
0
yd
1
yd
2
. . .
⇤
·
2
6
6
6
4
ufuture(t)
ufuture(t 1)
ufuture(t 2)
.
.
.
3
7
7
7
5
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Slide 35 text
y4
y2
y1 y3 y5
y6
y7
u1 = u2 = · · · = 0
u0 = 1
x0 =0
y0
Now what if we had the impulse response recorded in our data-library?
⇥
g0 g1 g2 . . .
⇤
=
⇥
yd
0
yd
1
yd
2
. . .
⇤
! dynamic matrix control
(Shell, 1970s): predictive
control from raw data
yfuture(t) =
⇥
yd
0
yd
1
yd
2
. . .
⇤
·
2
6
6
6
4
ufuture(t)
ufuture(t 1)
ufuture(t 2)
.
.
.
3
7
7
7
5
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Slide 36
Slide 36 text
y4
y2
y1 y3 y5
y6
y7
u1 = u2 = · · · = 0
u0 = 1
x0 =0
y0
Now what if we had the impulse response recorded in our data-library?
⇥
g0 g1 g2 . . .
⇤
=
⇥
yd
0
yd
1
yd
2
. . .
⇤
! dynamic matrix control
(Shell, 1970s): predictive
control from raw data
yfuture(t) =
⇥
yd
0
yd
1
yd
2
. . .
⇤
·
2
6
6
6
4
ufuture(t)
ufuture(t 1)
ufuture(t 2)
.
.
.
3
7
7
7
5
today : arbitrary, finite, & corrupted data, ...stochastic & nonlinear ?
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Today’s menu
1. behavioral system theory: fundamental lemma
2. DeePC : data-enabled predictive control
3. robustification via salient regularizations
4. cases studies from wind & power systems
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+ tomatoes
Slide 38
Slide 38 text
Today’s menu
1. behavioral system theory: fundamental lemma
2. DeePC : data-enabled predictive control
3. robustification via salient regularizations
4. cases studies from wind & power systems
blooming literature (2-3 ArXiv / week)
9/53
Slide 39
Slide 39 text
Today’s menu
1. behavioral system theory: fundamental lemma
2. DeePC : data-enabled predictive control
3. robustification via salient regularizations
4. cases studies from wind & power systems
blooming literature (2-3 ArXiv / week)
! tutorial [link] to get started
• [link] to graduate school material
• [link] to survey
• [link] to related bachelor lecture
• [link] to related publications
DATA-DRIVEN CONTROL BASED ON BEHAVIORAL APPROACH:
FROM THEORY TO APPLICATIONS IN POWER SYSTEMS
Ivan Markovsky, Linbin Huang, and Florian Dörfler
I. Markovsky is with ICREA, Pg. Lluis Companys 23, Barcelona, and CIMNE, Gran Capitàn, Barcelona, Spain
(e-mail: [email protected]),
L. Huang and F. Dörfler are with the Automatic Control Laboratory, ETH Zürich, 8092 Zürich, Switzerland (e-mails:
[email protected], dorfl[email protected]).
modeling). Modeling using observed data, possibly incorporating
some prior knowledge from the physical laws (that is, black-box
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Organization of this lecture
• I will teach the basics & provide pointers to more sophisticated
research material ! study cutting-edge papers yourself
• it’s a school: so we will spend time on the board ! take notes
• We teach this material also in the ETH Z¨
urich bachelor & have
plenty of background material + implementation experience
! please reach out to me or Saverio if you need anything
• we will take a break after 90 minutes ! co ee ,
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Preview
complex 4-area power system:
large (n=208), few sensors (8),
nonlinear, noisy, sti , input
constraints, & decentralized
control specifications
control objective: oscillation
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11/53
Slide 42
Slide 42 text
Preview
complex 4-area power system:
large (n=208), few sensors (8),
nonlinear, noisy, sti , input
constraints, & decentralized
control specifications
control objective: oscillation
damping
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time (s)
uncontrolled flow (p.u.)
11/53
Slide 43
Slide 43 text
Preview
complex 4-area power system:
large (n=208), few sensors (8),
nonlinear, noisy, sti , input
constraints, & decentralized
control specifications
control objective: oscillation
damping without a model
(grid has many owners, models are
proprietary, operation in flux, ...)
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! " #! #
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time (s)
uncontrolled flow (p.u.)
11/53
Slide 44
Slide 44 text
Preview
complex 4-area power system:
large (n=208), few sensors (8),
nonlinear, noisy, sti , input
constraints, & decentralized
control specifications
control objective: oscillation
damping without a model
(grid has many owners, models are
proprietary, operation in flux, ...)
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234*#$-5$.5%
234*#$.5$/5$
234*#$.5$/5%
234*#$/5$0
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7;4:);<#!3=4+<>
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2;+B#& 2;+B#'
control
control
! " #! #
!&!
!&$
!&'
!&(
10
time (s)
uncontrolled flow (p.u.)
collect data control
tie line flow (p.u.)
!"#$%&'(
! " #! #" $! $" %!
!&!
!&$
!&'
!&(
11/53
Slide 45
Slide 45 text
Preview
complex 4-area power system:
large (n=208), few sensors (8),
nonlinear, noisy, sti , input
constraints, & decentralized
control specifications
control objective: oscillation
damping without a model
(grid has many owners, models are
proprietary, operation in flux, ...)
!"#$
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()*+#$ ()*+#%
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control
control
! " #! #
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10
time (s)
uncontrolled flow (p.u.)
collect data control
tie line flow (p.u.)
!"#$%&'(
! " #! #" $! $" %!
!&!
!&$
!&'
!&(
seek a method that works
reliably, can be e ciently
implemented, & certifiable
! automating ourselves
11/53
Slide 46
Slide 46 text
Reality check: black magic or hoax ?
surely, nobody would put apply such a shaky data-driven method
• on the world’s most complex engineered system (the electric grid),
• using the world’s biggest actuators (Gigawatt-sized HVDC links),
• and subject to real-time, safety, stability, constraints ...right?
12/53
Slide 47
Slide 47 text
Reality check: black magic or hoax ?
surely, nobody would put apply such a shaky data-driven method
• on the world’s most complex engineered system (the electric grid),
• using the world’s biggest actuators (Gigawatt-sized HVDC links),
• and subject to real-time, safety, stability, constraints ...right?
!"#$%&'()'(%#(*%+,-$'#(.
%
/% 0123% 21)4'5"*% #% 6"$7% 8#6-1$#),"% $"6'"9% -8% 7-1$% :#:"$% ;<=% 9>'?>% /% )",'"6"% ?-1,*% )"% -8%
'4:-$3#(?"%3-%-1$%9-$@%#3%A'3#?>'%B-9"$%C$'*2D%E"%*-%>#6"%%;<=%$"F1'$"%-GH,'("%31('(I%3>#3%%;<=%
%
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2723"4% 4-*",2% -(% 3>"% '(3"$#$"#% -2?',,#J-(% :$-),"4D% L-1,*% 2-% 2-4"% ?-*"% )"% 4#*"% #6#',#),"%
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12/53
Slide 48
Slide 48 text
Reality check: black magic or hoax ?
surely, nobody would put apply such a shaky data-driven method
• on the world’s most complex engineered system (the electric grid),
• using the world’s biggest actuators (Gigawatt-sized HVDC links),
• and subject to real-time, safety, stability, constraints ...right?
!"#$%&'()'(%#(*%+,-$'#(.
%
/% 0123% 21)4'5"*% #% 6"$7% 8#6-1$#),"% $"6'"9% -8% 7-1$% :#:"$% ;<=% 9>'?>% /% )",'"6"% ?-1,*% )"% -8%
'4:-$3#(?"%3-%-1$%9-$@%#3%A'3#?>'%B-9"$%C$'*2D%E"%*-%>#6"%%;<=%$"F1'$"%-GH,'("%31('(I%3>#3%%;<=%
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%
/8%:-22'),"%/%9-1,*%,'@"%3-%3$7%3>"%*"?"(3$#,'K"*%!""BL%#::$-#?>%9'3>%-1$%4-$"%*"3#',"*%AM!L%
2723"4% 4-*",2% -(% 3>"% '(3"$#$"#% -2?',,#J-(% :$-),"4D% L-1,*% 2-% 2-4"% ?-*"% )"% 4#*"% #6#',#),"%
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2-Q9#$"%:,#R-$4
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2"(*'(I%-1$%?-*"
12/53
Slide 49
Slide 49 text
Reality check: black magic or hoax ?
surely, nobody would put apply such a shaky data-driven method
• on the world’s most complex engineered system (the electric grid),
• using the world’s biggest actuators (Gigawatt-sized HVDC links),
• and subject to real-time, safety, stability, constraints ...right?
!"#$%&'()'(%#(*%+,-$'#(.
%
/% 0123% 21)4'5"*% #% 6"$7% 8#6-1$#),"% $"6'"9% -8% 7-1$% :#:"$% ;<=% 9>'?>% /% )",'"6"% ?-1,*% )"% -8%
'4:-$3#(?"%3-%-1$%9-$@%#3%A'3#?>'%B-9"$%C$'*2D%E"%*-%>#6"%%;<=%$"F1'$"%-GH,'("%31('(I%3>#3%%;<=%
%
?-4'22'2'-('(I%"(I'(""$%?#(%*-%-(%>'2%-9(%%;<=%%#(%#*#:J6"%#::$-#?>%9-1,*%)"%6"$7%'(3"$"2J(ID
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/8%:-22'),"%/%9-1,*%,'@"%3-%3$7%3>"%*"?"(3$#,'K"*%!""BL%#::$-#?>%9'3>%-1$%4-$"%*"3#',"*%AM!L%
2723"4% 4-*",2% -(% 3>"% '(3"$#$"#% -2?',,#J-(% :$-),"4D% L-1,*% 2-% 2-4"% ?-*"% )"% 4#*"% #6#',#),"%
%
;<=%%N%E-1,*%7-1%)"%'(3"$"23"*%'(%9-$@'(I%3-I"3>"$%3-%*-%21?>%#%*"4-(23$#J-(%N%%;<=
%/3%9-$@2O%<%"6"(%
-(%#(%"(J$",7%
*'G"$"(3%4-*",%P%
2-Q9#$"%:,#R-$4
<%8"9%*#72%#Q"$%
2"(*'(I%-1$%?-*"
at least someone believes that our method is practically useful ...
12/53
Slide 50
Slide 50 text
LTI system representations
13/53
1 exogeneous in put
· ARX : y(+ + 2) + 2y(+ +
1) + 3y(t) = 4u(x)
11
auto-regressive
·
ARX-state space :
x() =
[ii]
x(t+
1) =
[-
=
]x(t) + [i]u(
·
ARX-
>
transfer
function :
y(H) =
[n0]x(t)
Y(z) =
2 ,
34(2)
us these are all parametric kernel representations
Slide 51
Slide 51 text
14/53
Iy (+ + 2) + 2y(+ +
1) + 3y(t) = 4u(t)
(time Shift 3 :
y(t) =
y(++ 1)
(2 +
28 + 3
,
4][37 = 0
~ bervel representation"
Slide 52
Slide 52 text
Behavioral view on dynamical systems
Definition: A discrete-time dynamical
system is a 3-tuple (Z 0, W, B) where
(i) Z 0
is the discrete-time axis,
(ii) W is the signal space, &
(iii) B ✓ WZ 0 is the behavior.
9
>
>
=
>
>
;
B is the set of
all trajectories
15/53
Yet
of all discrete time series W
Slide 53
Slide 53 text
Behavioral view on dynamical systems
Definition: A discrete-time dynamical
system is a 3-tuple (Z 0, W, B) where
(i) Z 0
is the discrete-time axis,
(ii) W is the signal space, &
(iii) B ✓ WZ 0 is the behavior.
9
>
>
=
>
>
;
B is the set of
all trajectories
Definition: The dynamical system (Z 0, W, B) is
(i) linear if W is a vector space & B is a subspace of WZ 0
(ii) & time-invariant if B ✓ B, where wt = wt+1
.
15/53
Slide 54
Slide 54 text
Behavioral view on dynamical systems
Definition: A discrete-time dynamical
system is a 3-tuple (Z 0, W, B) where
(i) Z 0
is the discrete-time axis,
(ii) W is the signal space, &
(iii) B ✓ WZ 0 is the behavior.
9
>
>
=
>
>
;
B is the set of
all trajectories
Definition: The dynamical system (Z 0, W, B) is
(i) linear if W is a vector space & B is a subspace of WZ 0
(ii) & time-invariant if B ✓ B, where wt = wt+1
.
y
u
15/53
Slide 55
Slide 55 text
Behavioral view on dynamical systems
Definition: A discrete-time dynamical
system is a 3-tuple (Z 0, W, B) where
(i) Z 0
is the discrete-time axis,
(ii) W is the signal space, &
(iii) B ✓ WZ 0 is the behavior.
9
>
>
=
>
>
;
B is the set of
all trajectories
Definition: The dynamical system (Z 0, W, B) is
(i) linear if W is a vector space & B is a subspace of WZ 0
(ii) & time-invariant if B ✓ B, where wt = wt+1
.
LTI system = shift-invariant subspace of trajectory space
! abstract perspective suited for data-driven control
y
u
15/53
Slide 56
Slide 56 text
Properties of the LTI trajectory space
16/53
/EIRY /ERM
·
model x (t+
1) =
Ax ( + ) + Bult X (0) =
Xini
X (1) = AXini + Buld
EM/YH=CX +
Do
X(2) =
#Xini + ABu(0
: + Bu(d
y(t) = (A
+
xin + A Bu(t) + Du(t)
in
vector notation :
I IB
D
le ( I+
y(T) CAT
u m
extended It :
extended
observabilitmatix Of impulseresponse
Slide 57
Slide 57 text
17/53
compactly: y =
Of Xini + 2 - u
·
observability : Xini can be reconstructed from (y ,
4)
from E) rank 0 = n
·
the smallest integer e so
that Or has rankn is
called the lg of the system : Sh JovaSISOyes is
= given past data Mini =
[u] and
gi
=> Xini can
be uniquely reconstructed E> Tini ? C
Slide 58
Slide 58 text
18/53
dimension of the L
TI
trajectory space
& Xini ER"
,
what is
the dimension of /ER"
It =
(i) I *I
U
u m ⊥
colem
1
pm .
T
/
column has always has
ranh n for Tze rank m .
T
E dimension o [Y] = m .
T
+ 4
for is
Slide 59
Slide 59 text
LTI systems & matrix time series
foundation of subspace system identification & signal recovery algorithms
u(t)
t
u4
u2
u1 u3
u5 u6
u7
y(t)
t
y4
y2
y1
y3
y5
y6
y7
19/53
Slide 60
Slide 60 text
LTI systems & matrix time series
foundation of subspace system identification & signal recovery algorithms
u(t)
t
u4
u2
u1 u3
u5 u6
u7
y(t)
t
y4
y2
y1
y3
y5
y6
y7
u(t), y(t) satisfy LTI
di erence equation
b0ut+b1ut+1+. . .+bnut+n+
a0yt+a1yt+1+. . .+anyt+n = 0
(ARX / kernel representation)
19/53
Slide 61
Slide 61 text
LTI systems & matrix time series
foundation of subspace system identification & signal recovery algorithms
u(t)
t
u4
u2
u1 u3
u5 u6
u7
y(t)
t
y4
y2
y1
y3
y5
y6
y7
u(t), y(t) satisfy LTI
di erence equation
b0ut+b1ut+1+. . .+bnut+n+
a0yt+a1yt+1+. . .+anyt+n = 0
(ARX / kernel representation)
)
[ 0 b0 a0 b1 a1 ... bn an 0 ] in left nullspace
of trajectory matrix (collected data)
H
⇣
ud
yd
⌘
=
2
6
6
6
6
6
6
6
4
ud
1,1
yd
1,1
!
ud
1,2
yd
1,2
!
ud
1,3
yd
1,3
!
...
ud
2,1
yd
2,1
!
ud
2,2
yd
2,2
!
ud
2,3
yd
2,3
!
...
.
.
.
.
.
.
.
.
.
.
.
.
ud
T,1
yd
T,1
!
ud
T,2
yd
T,2
!
ud
T,3
yd
T,3
!
...
3
7
7
7
7
7
7
7
5
| {z }
1st experiment
| {z }
2nd
| {z }
3rd ...
19/53
Slide 62
Slide 62 text
LTI systems & matrix time series
foundation of subspace system identification & signal recovery algorithms
u(t)
t
u4
u2
u1 u3
u5 u6
u7
y(t)
t
y4
y2
y1
y3
y5
y6
y7
u(t), y(t) satisfy LTI
di erence equation
b0ut+b1ut+1+. . .+bnut+n+
a0yt+a1yt+1+. . .+anyt+n = 0
(ARX / kernel representation)
(
under assumptions
)
[ 0 b0 a0 b1 a1 ... bn an 0 ] in left nullspace
of trajectory matrix (collected data)
H
⇣
ud
yd
⌘
=
2
6
6
6
6
6
6
6
4
ud
1,1
yd
1,1
!
ud
1,2
yd
1,2
!
ud
1,3
yd
1,3
!
...
ud
2,1
yd
2,1
!
ud
2,2
yd
2,2
!
ud
2,3
yd
2,3
!
...
.
.
.
.
.
.
.
.
.
.
.
.
ud
T,1
yd
T,1
!
ud
T,2
yd
T,2
!
ud
T,3
yd
T,3
!
...
3
7
7
7
7
7
7
7
5
| {z }
1st experiment
| {z }
2nd
| {z }
3rd ...
19/53
Slide 63
Slide 63 text
Fundamental Lemma
u(t)
t
u4
u2
u1 u3
u5 u6
u7
y(t)
t
y4
y2
y1
y3
y5
y6
y7
Given: data
⇣
ud
i
yd
i
⌘
2 Rm+p
20/53
Slide 64
Slide 64 text
Fundamental Lemma
u(t)
t
u4
u2
u1 u3
u5 u6
u7
y(t)
t
y4
y2
y1
y3
y5
y6
y7
Given: data
⇣
ud
i
yd
i
⌘
2 Rm+p & LTI complexity parameters
⇢
lag `
order n
20/53
Slide 65
Slide 65 text
Fundamental Lemma
u(t)
t
u4
u2
u1 u3
u5 u6
u7
y(t)
t
y4
y2
y1
y3
y5
y6
y7
Given: data
⇣
ud
i
yd
i
⌘
2 Rm+p & LTI complexity parameters
⇢
lag `
order n
set of all T-length trajectories =
n
(u, y) 2 R(m+p)T : 9x 2 RnT s.t.
x+ = Ax + Bu , y = Cx + Du
o
| {z }
parametric state-space model
20/53
Slide 66
Slide 66 text
Fundamental Lemma
u(t)
t
u4
u2
u1 u3
u5 u6
u7
y(t)
t
y4
y2
y1
y3
y5
y6
y7
Given: data
⇣
ud
i
yd
i
⌘
2 Rm+p & LTI complexity parameters
⇢
lag `
order n
set of all T-length trajectories =
n
(u, y) 2 R(m+p)T : 9x 2 RnT s.t.
x+ = Ax + Bu , y = Cx + Du
o
| {z } | {z }
parametric state-space model raw data (every column is an experiment)
colspan
2
6
6
6
6
6
6
6
6
4
ud
1,1
yd
1,1
!
ud
1,2
yd
1,2
!
ud
1,3
yd
1,3
!
...
ud
2,1
yd
2,1
!
ud
2,2
yd
2,2
!
ud
2,3
yd
2,3
!
...
.
.
.
.
.
.
.
.
.
.
.
.
ud
T,1
yd
T,1
!
ud
T,2
yd
T,2
!
ud
T,3
yd
T,3
!
...
3
7
7
7
7
7
7
7
7
5
20/53
Slide 67
Slide 67 text
Fundamental Lemma
u(t)
t
u4
u2
u1 u3
u5 u6
u7
y(t)
t
y4
y2
y1
y3
y5
y6
y7
Given: data
⇣
ud
i
yd
i
⌘
2 Rm+p & LTI complexity parameters
⇢
lag `
order n
set of all T-length trajectories =
n
(u, y) 2 R(m+p)T : 9x 2 RnT s.t.
x+ = Ax + Bu , y = Cx + Du
o
| {z } | {z }
parametric state-space model raw data (every column is an experiment)
colspan
2
6
6
6
6
6
6
6
6
4
ud
1,1
yd
1,1
!
ud
1,2
yd
1,2
!
ud
1,3
yd
1,3
!
...
ud
2,1
yd
2,1
!
ud
2,2
yd
2,2
!
ud
2,3
yd
2,3
!
...
.
.
.
.
.
.
.
.
.
.
.
.
ud
T,1
yd
T,1
!
ud
T,2
yd
T,2
!
ud
T,3
yd
T,3
!
...
3
7
7
7
7
7
7
7
7
5
if and only if the trajectory matrix has rank m · T + n for all T > `
20/53
Slide 68
Slide 68 text
set of all T-length trajectories =
n
(u, y) 2 R(m+p)T : 9x 2 RnT s.t.
x+ = Ax + Bu , y = Cx + Du
o
| {z } | {z }
parametric state-space model non-parametric model from raw data
colspan
2
6
6
6
6
6
6
6
6
4
ud
1,1
yd
1,1
!
ud
1,2
yd
1,2
!
ud
1,3
yd
1,3
!
...
ud
2,1
yd
2,1
!
ud
2,2
yd
2,2
!
ud
2,3
yd
2,3
!
...
.
.
.
.
.
.
.
.
.
.
.
.
ud
T,1
yd
T,1
!
ud
T,2
yd
T,2
!
ud
T,3
yd
T,3
!
...
3
7
7
7
7
7
7
7
7
5
21/53
Slide 69
Slide 69 text
set of all T-length trajectories =
n
(u, y) 2 R(m+p)T : 9x 2 RnT s.t.
x+ = Ax + Bu , y = Cx + Du
o
| {z } | {z }
parametric state-space model non-parametric model from raw data
colspan
2
6
6
6
6
6
6
6
6
4
ud
1,1
yd
1,1
!
ud
1,2
yd
1,2
!
ud
1,3
yd
1,3
!
...
ud
2,1
yd
2,1
!
ud
2,2
yd
2,2
!
ud
2,3
yd
2,3
!
...
.
.
.
.
.
.
.
.
.
.
.
.
ud
T,1
yd
T,1
!
ud
T,2
yd
T,2
!
ud
T,3
yd
T,3
!
...
3
7
7
7
7
7
7
7
7
5
all trajectories constructible from finitely many previous trajectories
21/53
Slide 70
Slide 70 text
set of all T-length trajectories =
n
(u, y) 2 R(m+p)T : 9x 2 RnT s.t.
x+ = Ax + Bu , y = Cx + Du
o
| {z } | {z }
parametric state-space model non-parametric model from raw data
colspan
2
6
6
6
6
6
6
6
6
4
ud
1,1
yd
1,1
!
ud
1,2
yd
1,2
!
ud
1,3
yd
1,3
!
...
ud
2,1
yd
2,1
!
ud
2,2
yd
2,2
!
ud
2,3
yd
2,3
!
...
.
.
.
.
.
.
.
.
.
.
.
.
ud
T,1
yd
T,1
!
ud
T,2
yd
T,2
!
ud
T,3
yd
T,3
!
...
3
7
7
7
7
7
7
7
7
5
all trajectories constructible from finitely many previous trajectories
• standing on the shoulders of giants:
classic Willems’ result was only “if” &
required further assumptions: Hankel,
persistency of excitation, controllability
21/53
Slide 71
Slide 71 text
set of all T-length trajectories =
n
(u, y) 2 R(m+p)T : 9x 2 RnT s.t.
x+ = Ax + Bu , y = Cx + Du
o
| {z } | {z }
parametric state-space model non-parametric model from raw data
colspan
2
6
6
6
6
6
6
6
6
4
ud
1,1
yd
1,1
!
ud
1,2
yd
1,2
!
ud
1,3
yd
1,3
!
...
ud
2,1
yd
2,1
!
ud
2,2
yd
2,2
!
ud
2,3
yd
2,3
!
...
.
.
.
.
.
.
.
.
.
.
.
.
ud
T,1
yd
T,1
!
ud
T,2
yd
T,2
!
ud
T,3
yd
T,3
!
...
3
7
7
7
7
7
7
7
7
5
all trajectories constructible from finitely many previous trajectories
• standing on the shoulders of giants:
classic Willems’ result was only “if” &
required further assumptions: Hankel,
persistency of excitation, controllability
• terminology fundamental is justified : motion primitives, subspace SysID,
dictionary learning, (E)DMD, ... all implicitly rely on this equivalence
21/53
Slide 72
Slide 72 text
set of all T-length trajectories =
n
(u, y) 2 R(m+p)T : 9x 2 RnT s.t.
x+ = Ax + Bu , y = Cx + Du
o
| {z } | {z }
parametric state-space model non-parametric model from raw data
colspan
2
6
6
6
6
6
6
6
6
4
ud
1,1
yd
1,1
!
ud
1,2
yd
1,2
!
ud
1,3
yd
1,3
!
...
ud
2,1
yd
2,1
!
ud
2,2
yd
2,2
!
ud
2,3
yd
2,3
!
...
.
.
.
.
.
.
.
.
.
.
.
.
ud
T,1
yd
T,1
!
ud
T,2
yd
T,2
!
ud
T,3
yd
T,3
!
...
3
7
7
7
7
7
7
7
7
5
all trajectories constructible from finitely many previous trajectories
• standing on the shoulders of giants:
classic Willems’ result was only “if” &
required further assumptions: Hankel,
persistency of excitation, controllability
• terminology fundamental is justified : motion primitives, subspace SysID,
dictionary learning, (E)DMD, ... all implicitly rely on this equivalence
• many recent extensions to other system classes (bi-linear, descriptor,
LPV, delay, Volterra series, Wiener-Hammerstein, ...), other matrix
data structures (mosaic Hankel, Page, ...), & other proof methods
21/53
Slide 73
Slide 73 text
Input design for Fundamental Lemma
u(t)
t
u4
u2
u1 u3
u5 u6
u7
y(t)
t
y4
y2
y1
y3
y5
y6
y7
22/53
Slide 74
Slide 74 text
Input design for Fundamental Lemma
u(t)
t
u4
u2
u1 u3
u5 u6
u7
y(t)
t
y4
y2
y1
y3
y5
y6
y7
Definition: The data signal ud 2 RmTd of length Td
is persistently
exciting of order T if the Hankel matrix
2
4
u1
··· uTd T +1
.
.
.
...
.
.
.
uT
··· uTd
3
5 is of full rank.
22/53
↑
UT
+1 -
Slide 75
Slide 75 text
Input design for Fundamental Lemma
u(t)
t
u4
u2
u1 u3
u5 u6
u7
y(t)
t
y4
y2
y1
y3
y5
y6
y7
Definition: The data signal ud 2 RmTd of length Td
is persistently
exciting of order T if the Hankel matrix
2
4
u1
··· uTd T +1
.
.
.
...
.
.
.
uT
··· uTd
3
5 is of full rank.
22/53
m .
T
for full rank : Tn -T + mT => To is sufficiently
Slide 76
Slide 76 text
Input design for Fundamental Lemma
u(t)
t
u4
u2
u1 u3
u5 u6
u7
y(t)
t
y4
y2
y1
y3
y5
y6
y7
Definition: The data signal ud 2 RmTd of length Td
is persistently
exciting of order T if the Hankel matrix
2
4
u1
··· uTd T +1
.
.
.
...
.
.
.
uT
··· uTd
3
5 is of full rank.
Input design [Willems et al, ’05]: Controllable LTI system & persistently
exciting input ud of order T + n =) rank
⇣
H
⇣
ud
yd
⌘⌘
= mT + n.
22/53
Slide 77
Slide 77 text
Data matrix structures & preprocessing
23/53
· trajectory mix
H1w()
-
Spinar/perl-
]
requires independent experiment
·
page matrix = HIw) =
[W Wi Wat
requires one long experiment
· Hankel matrix =
H(w) =
[W W W. .
requires one short experiment
. . .
or
any combinations...
Slide 78
Slide 78 text
24/53
Pre-procession of noisy data :
if wh is noisy,
then all of the above matrices have full ranh
.
low-rank-preprocessing: min 11-w9
=
find the closest i
ranh (H(w) = mL + n
low-rank data
=> standard solution is to take an SVD of H(w/
and only keep the largest mi + n singular values
= is optimal if the matrix is unstructured
. . .
but does not apply time series are correlated
Slide 79
Slide 79 text
Bird’s view & today’s sample path
through the accelerating literature
Fundamental
Lemma [Willems,
Rapisarda, &
Markovsky ’05]
subspace
intersection
methods
[Moonen et al., ’89]
PE in linear
systems
[Green & Moore, ’86]
many recent
variations &
extensions
[van Waarde et al., ’20]
generalized low-
rank version
[Markovsky
& Dörfler, ’20]
deterministic
data-driven
control [Markovsky
& Rapisarda, ’08]
data-driven
control of linear
systems
[Persis & Tesi, ’19]
regularizations
& MPC scenario
[Coulson et al., ’19]
data
informativity
[van Waarde et al., ’20]
LFT formulation
[Berberich et al., ’20]
…
?
explicit
implicit
non-control
applications:
e.g., estimation.
filtering, & SysID stabilization of
nonlinear
systems
[Persis & Tesi, ’21]
…
robust stability
& recursive
feasibility
[Berberich et al., ’20]
(distributional)
robustness
[Coulson et al., ’20,
Huang et al., ’21]
regularizer from
relaxed SysID
[Dörfler et al., ’21]
…
… …
subspace
predictive
control
[Favoreel et al., ’99]
subspace
methods
[Breschi, Chiuso, &
Formention ’22]
instrumental
variables
[Wingerden et al., ’22]
1980s 2005 today
25/53
Slide 80
Slide 80 text
Output Model Predictive Control (MPC)
minimize
u, x, y
Tfuture
X
k=1
kyk rk
k2
Q
+ kuk
k2
R
subject to xk+1 = Axk + Buk
yk = Cxk + Duk
8k 2 {1, . . . , Tfuture
}
uk
2 U
yk
2 Y
8k 2 {1, . . . , Tfuture
}
quadratic cost with
R 0, Q ⌫ 0 & ref. r
model for prediction
with k 2 [1, Tfuture]
hard operational or
safety constraints
26/53
Slide 81
Slide 81 text
Output Model Predictive Control (MPC)
minimize
u, x, y
Tfuture
X
k=1
kyk rk
k2
Q
+ kuk
k2
R
subject to xk+1 = Axk + Buk
yk = Cxk + Duk
8k 2 {1, . . . , Tfuture
}
xk+1 = Axk + Buk
yk = Cxk + Duk
8k 2 { Tini 1, . . . , 0}
uk
2 U
yk
2 Y
8k 2 {1, . . . , Tfuture
}
quadratic cost with
R 0, Q ⌫ 0 & ref. r
model for prediction
with k 2 [1, Tfuture]
model for estimation
with k 2 [ Tini 1, 0] &
Tini
lag (many flavors)
hard operational or
safety constraints
26/53
Slide 82
Slide 82 text
Output Model Predictive Control (MPC)
minimize
u, x, y
Tfuture
X
k=1
kyk rk
k2
Q
+ kuk
k2
R
subject to xk+1 = Axk + Buk
yk = Cxk + Duk
8k 2 {1, . . . , Tfuture
}
xk+1 = Axk + Buk
yk = Cxk + Duk
8k 2 { Tini 1, . . . , 0}
uk
2 U
yk
2 Y
8k 2 {1, . . . , Tfuture
}
quadratic cost with
R 0, Q ⌫ 0 & ref. r
model for prediction
with k 2 [1, Tfuture]
model for estimation
with k 2 [ Tini 1, 0] &
Tini
lag (many flavors)
hard operational or
safety constraints
“[MPC] has perhaps too little system
theory and too much
brute force
– Willems ’07
26/53
Slide 83
Slide 83 text
Output Model Predictive Control (MPC)
minimize
u, x, y
Tfuture
X
k=1
kyk rk
k2
Q
+ kuk
k2
R
subject to xk+1 = Axk + Buk
yk = Cxk + Duk
8k 2 {1, . . . , Tfuture
}
xk+1 = Axk + Buk
yk = Cxk + Duk
8k 2 { Tini 1, . . . , 0}
uk
2 U
yk
2 Y
8k 2 {1, . . . , Tfuture
}
quadratic cost with
R 0, Q ⌫ 0 & ref. r
model for prediction
with k 2 [1, Tfuture]
model for estimation
with k 2 [ Tini 1, 0] &
Tini
lag (many flavors)
hard operational or
safety constraints
“[MPC] has perhaps too little system
theory and too much
brute force
[...], but
MPC is an area where all aspects of the
field [...] are in synergy.” – Willems ’07
26/53
Slide 84
Slide 84 text
Output Model Predictive Control (MPC)
minimize
u, x, y
Tfuture
X
k=1
kyk rk
k2
Q
+ kuk
k2
R
subject to xk+1 = Axk + Buk
yk = Cxk + Duk
8k 2 {1, . . . , Tfuture
}
xk+1 = Axk + Buk
yk = Cxk + Duk
8k 2 { Tini 1, . . . , 0}
uk
2 U
yk
2 Y
8k 2 {1, . . . , Tfuture
}
quadratic cost with
R 0, Q ⌫ 0 & ref. r
model for prediction
with k 2 [1, Tfuture]
model for estimation
with k 2 [ Tini 1, 0] &
Tini
lag (many flavors)
hard operational or
safety constraints
“[MPC] has perhaps too little system
theory and too much
brute force
[...], but
MPC is an area where all aspects of the
field [...] are in synergy.” – Willems ’07
Elegance aside, for an LTI
plant, deterministic, & with
known model, MPC is the
gold standard of control.
26/53
Slide 85
Slide 85 text
Data-enabled Predictive Control (DeePC)
27/53
Slide 86
Slide 86 text
Data-enabled Predictive Control (DeePC)
minimize
g, u, y
Tfuture
X
k=1
kyk rk
k2
Q
+ kuk
k2
R
subject to H
⇣
ud
yd
⌘
· g =
2
6
6
4
uini
yini
u
y
3
7
7
5
uk
2 U
yk
2 Y
8k 2 {1, . . . , Tfuture
}
quadratic cost with
R 0, Q ⌫ 0 & ref. r
non-parametric
model for prediction
and estimation
hard operational or
safety constraints
• real-time measurements (uini, yini) for estimation
• trajectory matrix H
⇣
ud
yd
⌘
from past
experimental data
updated online
collected o ine
(could be adapted online)
27/53
oflength Tini-e
Slide 87
Slide 87 text
Data-enabled Predictive Control (DeePC)
minimize
g, u, y
Tfuture
X
k=1
kyk rk
k2
Q
+ kuk
k2
R
subject to H
⇣
ud
yd
⌘
· g =
2
6
6
4
uini
yini
u
y
3
7
7
5
uk
2 U
yk
2 Y
8k 2 {1, . . . , Tfuture
}
quadratic cost with
R 0, Q ⌫ 0 & ref. r
non-parametric
model for prediction
and estimation
hard operational or
safety constraints
• real-time measurements (uini, yini) for estimation
• trajectory matrix H
⇣
ud
yd
⌘
from past
experimental data
updated online
collected o ine
(could be adapted online)
! equivalent to MPC in deterministic LTI case ...
but needs to be robustified in case of noise / nonlinearity ! 27/53
#
Slide 88
Slide 88 text
Regularizations to make it work
minimize
g, u, y
Tfuture
X
k=1
kyk rk
k2
Q
+ kuk
k2
R
subject to H
⇣
ud
yd
⌘
· g =
2
6
6
4
uini
yini
u
y
3
7
7
5
uk
2 U
yk
2 Y
8k 2 {1, . . . , Tfuture
}
28/53
Slide 89
Slide 89 text
Regularizations to make it work
minimize
g, u, y,
Tfuture
X
k=1
kyk rk
k2
Q
+ kuk
k2
R
+ y
k kp
subject to H
⇣
ud
yd
⌘
· g =
2
6
6
4
uini
yini
u
y
3
7
7
5 +
2
6
6
4
0
0
0
3
7
7
5
uk
2 U
yk
2 Y
8k 2 {1, . . . , Tfuture
}
measurement noise
! infeasible yini
estimate
! estimation slack
! moving-horizon
least-square filter
28/53
Slide 90
Slide 90 text
Regularizations to make it work
minimize
g, u, y,
Tfuture
X
k=1
kyk rk
k2
Q
+ kuk
k2
R
+ y
k kp + gh(g)
subject to H
⇣
ud
yd
⌘
· g =
2
6
6
4
uini
yini
u
y
3
7
7
5 +
2
6
6
4
0
0
0
3
7
7
5
uk
2 U
yk
2 Y
8k 2 {1, . . . , Tfuture
}
measurement noise
! infeasible yini
estimate
! estimation slack
! moving-horizon
least-square filter
noisy or nonlinear
(o ine) data matrix
! any (u
y) feasible
! add regularizer h(g)
28/53
Slide 91
Slide 91 text
Regularizations to make it work
minimize
g, u, y,
Tfuture
X
k=1
kyk rk
k2
Q
+ kuk
k2
R
+ y
k kp + gh(g)
subject to H
⇣
ud
yd
⌘
· g =
2
6
6
4
uini
yini
u
y
3
7
7
5 +
2
6
6
4
0
0
0
3
7
7
5
uk
2 U
yk
2 Y
8k 2 {1, . . . , Tfuture
}
measurement noise
! infeasible yini
estimate
! estimation slack
! moving-horizon
least-square filter
noisy or nonlinear
(o ine) data matrix
! any (u
y) feasible
! add regularizer h(g)
Bayesian intuition: regularization , prior, e.g., h(g) = kgk1
sparsely
selects {trajectory matrix columns} = {motion primitives} ⇠ low-order basis
28/53
Slide 92
Slide 92 text
Regularizations to make it work
minimize
g, u, y,
Tfuture
X
k=1
kyk rk
k2
Q
+ kuk
k2
R
+ y
k kp + gh(g)
subject to H
⇣
ud
yd
⌘
· g =
2
6
6
4
uini
yini
u
y
3
7
7
5 +
2
6
6
4
0
0
0
3
7
7
5
uk
2 U
yk
2 Y
8k 2 {1, . . . , Tfuture
}
measurement noise
! infeasible yini
estimate
! estimation slack
! moving-horizon
least-square filter
noisy or nonlinear
(o ine) data matrix
! any (u
y) feasible
! add regularizer h(g)
Bayesian intuition: regularization , prior, e.g., h(g) = kgk1
sparsely
selects {trajectory matrix columns} = {motion primitives} ⇠ low-order basis
Robustness intuition: regularization , robustifies, e.g., in a simple case
28/53
Slide 93
Slide 93 text
Regularizations to make it work
minimize
g, u, y,
Tfuture
X
k=1
kyk rk
k2
Q
+ kuk
k2
R
+ y
k kp + gh(g)
subject to H
⇣
ud
yd
⌘
· g =
2
6
6
4
uini
yini
u
y
3
7
7
5 +
2
6
6
4
0
0
0
3
7
7
5
uk
2 U
yk
2 Y
8k 2 {1, . . . , Tfuture
}
measurement noise
! infeasible yini
estimate
! estimation slack
! moving-horizon
least-square filter
noisy or nonlinear
(o ine) data matrix
! any (u
y) feasible
! add regularizer h(g)
Bayesian intuition: regularization , prior, e.g., h(g) = kgk1
sparsely
selects {trajectory matrix columns} = {motion primitives} ⇠ low-order basis
Robustness intuition: regularization , robustifies, e.g., in a simple case
28/53
minmuxl(+
)x-3min is
llAx-b11 + 11XX1 =
min 11Ax-b1 + g1xI
* 10/118
Slide 94
Slide 94 text
regularization
m
incorporating priors
+ implicit SysID
Slide 95
Slide 95 text
Regularization = relaxing low-rank
approximation in pre-processing
minimizeu,y,g control cost u, y
subject to
u
y
= H
⇣
ˆ
u
ˆ
y
⌘
g
where
⇣
ˆ
u
ˆ
y
⌘
2 argmin
⇣
ˆ
u
ˆ
y
⌘ ⇣
ud
yd
⌘
subject to rank H
ˆ
u
ˆ
y
= mL + n
9
=
;
optimal control
9
=
;
low-rank approximation
29/53
Slide 96
Slide 96 text
Regularization = relaxing low-rank
approximation in pre-processing
minimizeu,y,g control cost u, y
subject to
u
y
= H
⇣
ˆ
u
ˆ
y
⌘
g
where
⇣
ˆ
u
ˆ
y
⌘
2 argmin
⇣
ˆ
u
ˆ
y
⌘ ⇣
ud
yd
⌘
subject to rank H
ˆ
u
ˆ
y
= mL + n
# sequence of convex relaxations #
minimizeu,y,g control cost u, y
subject to
u
y
= H
⇣
ˆ
u
ˆ
y
⌘
g
where
⇣
ˆ
u
ˆ
y
⌘
2 argmin
⇣
ˆ
u
ˆ
y
⌘ ⇣
ud
yd
⌘
subject to rank H
ˆ
u
ˆ
y
= mL + n
, kgk0
mL + n
9
=
;
optimal control
9
=
;
low-rank approximation
29/53
Slide 97
Slide 97 text
Regularization = relaxing low-rank
approximation in pre-processing
minimizeu,y,g control cost u, y
subject to
u
y
= H
⇣
ˆ
u
ˆ
y
⌘
g
where
⇣
ˆ
u
ˆ
y
⌘
2 argmin
⇣
ˆ
u
ˆ
y
⌘ ⇣
ud
yd
⌘
subject to rank H
ˆ
u
ˆ
y
= mL + n
# sequence of convex relaxations #
minimizeu,y,g control cost u, y
subject to
u
y
= H
⇣
ˆ
u
ˆ
y
⌘
g
where
⇣
ˆ
u
ˆ
y
⌘
2 argmin
⇣
ˆ
u
ˆ
y
⌘ ⇣
ud
yd
⌘
subject to rank H
ˆ
u
ˆ
y
= mL + n
, kgk0
mL + n
9
=
;
optimal control
9
=
;
low-rank approximation
29/53
Slide 98
Slide 98 text
Regularization = relaxing low-rank
approximation in pre-processing
minimizeu,y,g control cost u, y
subject to
u
y
= H
⇣
ˆ
u
ˆ
y
⌘
g
where
⇣
ˆ
u
ˆ
y
⌘
2 argmin
⇣
ˆ
u
ˆ
y
⌘ ⇣
ud
yd
⌘
subject to rank H
ˆ
u
ˆ
y
= mL + n
# sequence of convex relaxations #
minimizeu,y,g control cost u, y
subject to
u
y
= H
⇣
ˆ
u
ˆ
y
⌘
g
where
⇣
ˆ
u
ˆ
y
⌘
2 argmin
⇣
ˆ
u
ˆ
y
⌘ ⇣
ud
yd
⌘
subject to rank H
ˆ
u
ˆ
y
= mL + n
, kgk0
mL + n
=
⇣
ud
yd
⌘
9
=
;
optimal control
9
=
;
low-rank approximation
29/53
Slide 99
Slide 99 text
Regularization = relaxing low-rank
approximation in pre-processing
minimizeu,y,g control cost u, y
subject to
u
y
= H
⇣
ˆ
u
ˆ
y
⌘
g
where
⇣
ˆ
u
ˆ
y
⌘
2 argmin
⇣
ˆ
u
ˆ
y
⌘ ⇣
ud
yd
⌘
subject to rank H
ˆ
u
ˆ
y
= mL + n
# sequence of convex relaxations #
minimizeu,y,g control cost u, y
subject to
u
y
= H
⇣
ud
yd
⌘
g , kgk0
mL + n
9
=
;
optimal control
9
=
;
low-rank approximation
29/53
Slide 100
Slide 100 text
Regularization = relaxing low-rank
approximation in pre-processing
minimizeu,y,g control cost u, y
subject to
u
y
= H
⇣
ˆ
u
ˆ
y
⌘
g
where
⇣
ˆ
u
ˆ
y
⌘
2 argmin
⇣
ˆ
u
ˆ
y
⌘ ⇣
ud
yd
⌘
subject to rank H
ˆ
u
ˆ
y
= mL + n
# sequence of convex relaxations #
minimizeu,y,g control cost u, y
subject to
u
y
= H
⇣
ud
yd
⌘
g , kgk1
mL + n
9
=
;
optimal control
9
=
;
low-rank approximation
29/53
Slide 101
Slide 101 text
Regularization = relaxing low-rank
approximation in pre-processing
minimizeu,y,g control cost u, y
subject to
u
y
= H
⇣
ˆ
u
ˆ
y
⌘
g
where
⇣
ˆ
u
ˆ
y
⌘
2 argmin
⇣
ˆ
u
ˆ
y
⌘ ⇣
ud
yd
⌘
subject to rank H
ˆ
u
ˆ
y
= mL + n
# sequence of convex relaxations #
minimizeu,y,g control cost u, y + g
· kgk1
subject to
u
y
= H
⇣
ud
yd
⌘
g
9
=
;
optimal control
9
=
;
low-rank approximation
29/53
Slide 102
Slide 102 text
Regularization = relaxing low-rank
approximation in pre-processing
minimizeu,y,g control cost u, y
subject to
u
y
= H
⇣
ˆ
u
ˆ
y
⌘
g
where
⇣
ˆ
u
ˆ
y
⌘
2 argmin
⇣
ˆ
u
ˆ
y
⌘ ⇣
ud
yd
⌘
subject to rank H
ˆ
u
ˆ
y
= mL + n
# sequence of convex relaxations #
minimizeu,y,g control cost u, y + g
· kgk1
subject to
u
y
= H
⇣
ud
yd
⌘
g
`1
-regularization = relaxation of low-rank
approximation & smoothened order selection
9
=
;
optimal control
9
=
;
low-rank approximation
29/53
Slide 103
Slide 103 text
Regularization = relaxing low-rank
approximation in pre-processing
minimizeu,y,g control cost u, y
subject to
u
y
= H
⇣
ˆ
u
ˆ
y
⌘
g
where
⇣
ˆ
u
ˆ
y
⌘
2 argmin
⇣
ˆ
u
ˆ
y
⌘ ⇣
ud
yd
⌘
subject to rank H
ˆ
u
ˆ
y
= mL + n
# sequence of convex relaxations #
minimizeu,y,g control cost u, y + g
· kgk1
subject to
u
y
= H
⇣
ud
yd
⌘
g
`1
-regularization = relaxation of low-rank
approximation & smoothened order selection
9
=
;
optimal control
9
=
;
low-rank approximation
!"#$%&'"##()*#$+
realized closed-loop cost
g
29/53
i
To 104
Slide 104
Slide 104 text
Certainty-Equivalence Regularizer
30/53
ARX representation of predictor:
DeePC representation of predictor:
y =
Of Xini + 2
+
4
where Xini satisfies Jin=
Finixini
= Hog =g
+
Tini Kini
=>
y =
1 .
[ii] + "noise or
y = Yeg ,
where (ii)=
where K is learned from data
1 = arguin 114-1 [ ]/
or
y=
Y [YgY)+ Ygna
= y .
(i grouze
Gone (i)
to re-create
=> y =
y
,
/(ii) "SPC the model-based solutiona
we need to penalize grom
Slide 105
Slide 105 text
Regularization , reformulate subspace ID
! indirect SysID + control problem
minimizeu,y
control cost(u, y)
subject to y = K?
2
4
uini
yini
u
3
5
31/53
Slide 106
Slide 106 text
Regularization , reformulate subspace ID
partition data as in subspace ID:
H
⇣
ud
yd
⌘
⇠
2
6
6
4
Up
Yp
Uf
Yf
3
7
7
5
(m + p)Tini
(m + p)Tfuture
! indirect SysID + control problem
minimizeu,y
control cost(u, y)
subject to y = K?
2
4
uini
yini
u
3
5
31/53
Slide 107
Slide 107 text
Regularization , reformulate subspace ID
partition data as in subspace ID:
H
⇣
ud
yd
⌘
⇠
2
6
6
4
Up
Yp
Uf
Yf
3
7
7
5
(m + p)Tini
(m + p)Tfuture
! indirect SysID + control problem
minimizeu,y
control cost(u, y)
subject to y = K?
2
4
uini
yini
u
3
5
where K? = argmin
K
YF K
2
4
Up
Yp
Uf
3
5
31/53
Slide 108
Slide 108 text
Regularization , reformulate subspace ID
partition data as in subspace ID:
H
⇣
ud
yd
⌘
⇠
2
6
6
4
Up
Yp
Uf
Yf
3
7
7
5
(m + p)Tini
(m + p)Tfuture
ID of optimal multi-step predictor
as in SPC: K? = YF
Up
Yp
Uf
†
8
<
:
! indirect SysID + control problem
minimizeu,y
control cost(u, y)
subject to y = K?
2
4
uini
yini
u
3
5
where K? = argmin
K
YF K
2
4
Up
Yp
Uf
3
5
31/53
Slide 109
Slide 109 text
Regularization , reformulate subspace ID
partition data as in subspace ID:
H
⇣
ud
yd
⌘
⇠
2
6
6
4
Up
Yp
Uf
Yf
3
7
7
5
(m + p)Tini
(m + p)Tfuture
ID of optimal multi-step predictor
as in SPC: K? = YF
Up
Yp
Uf
†
8
<
:
! indirect SysID + control problem
minimizeu,y
control cost(u, y)
subject to y = K?
2
4
uini
yini
u
3
5
where K? = argmin
K
YF K
2
4
Up
Yp
Uf
3
5
The above is equivalent
to regularized DeePC minimizeg,u,y
control cost(u, y) + g
Proj
⇣
ud
yd
⌘
g
p
subject to H
⇣
ud
yd
⌘
· g =
2
6
6
4
uini
yini
u
y
3
7
7
5
31/53
Slide 110
Slide 110 text
Regularization , reformulate subspace ID
partition data as in subspace ID:
H
⇣
ud
yd
⌘
⇠
2
6
6
4
Up
Yp
Uf
Yf
3
7
7
5
(m + p)Tini
(m + p)Tfuture
ID of optimal multi-step predictor
as in SPC: K? = YF
Up
Yp
Uf
†
8
<
:
! indirect SysID + control problem
minimizeu,y
control cost(u, y)
subject to y = K?
2
4
uini
yini
u
3
5
where K? = argmin
K
YF K
2
4
Up
Yp
Uf
3
5
The above is equivalent
to regularized DeePC
where Proj
⇣
ud
yd
⌘
projects
orthogonal to ker
Up
Yp
Uf
minimizeg,u,y
control cost(u, y) + g
Proj
⇣
ud
yd
⌘
g
p
subject to H
⇣
ud
yd
⌘
· g =
2
6
6
4
uini
yini
u
y
3
7
7
5
31/53
Slide 111
Slide 111 text
Performance of regularizers applied
to a stochastic LTI system
kgkp
Proj
⇣
ud
yd
⌘
g
p
32/53
Slide 112
Slide 112 text
Performance of regularizers applied
to a stochastic LTI system
kgkp
Proj
⇣
ud
yd
⌘
g
p
Hanke-Raus
heuristic (often)
reveals
32/53
Slide 113
Slide 113 text
Case study: wind turbine
• detailed industrial model: 37 states &
highly nonlinear (abc $ dq, MPTT,
PLL, power specs, dynamics, etc.)
• turbine & grid model unknown to
commissioning engineer & operator
• weak grid + PLL + fault ! loss of sync
• disturbance to be rejected by DeePC 33/53
Slide 114
Slide 114 text
Case study: wind turbine
• detailed industrial model: 37 states &
highly nonlinear (abc $ dq, MPTT,
PLL, power specs, dynamics, etc.)
• turbine & grid model unknown to
commissioning engineer & operator
• weak grid + PLL + fault ! loss of sync
• disturbance to be rejected by DeePC
!"#"
$%&&'$#(%)
*(#+%,#-"!!(#(%)"&-$%)#.%&
%/$(&&"#(%)
%0/'.1'!
h(g) = kgk2
2
h(g) = kgk1
h(g) = Proj
⇣
ud
yd
⌘
g
2
2
2''34-"$#(1"#'!
2''34-"$#(1"#'!
33/53
Slide 115
Slide 115 text
Case study: wind turbine
• detailed industrial model: 37 states &
highly nonlinear (abc $ dq, MPTT,
PLL, power specs, dynamics, etc.)
• turbine & grid model unknown to
commissioning engineer & operator
• weak grid + PLL + fault ! loss of sync
• disturbance to be rejected by DeePC
!"#"
$%&&'$#(%)
*(#+%,#-"!!(#(%)"&-$%)#.%&
%/$(&&"#(%)
%0/'.1'!
h(g) = kgk2
2
h(g) = kgk1
h(g) = Proj
⇣
ud
yd
⌘
g
2
2
2''34-"$#(1"#'!
2''34-"$#(1"#'!
regularizer tuning h(g) = kgk2
2
h(g) = kgk1
h(g) = Proj
⇣
ud
yd
⌘
g
2
2
Hanke-Raus heuristic
33/53
Slide 116
Slide 116 text
Case study ++ : wind farm
SG 1
SG 2 SG 3
1
2 3
4
5 6
7 9
8
IEEE nine-bus system
wind farm
1
2
3
4
5
6
7
8
9
10
• high-fidelity models for turbines,
machines, & IEEE-9-bus system
• fast frequency response via
decentralized DeePC at turbines 34/53
Slide 117
Slide 117 text
Case study ++ : wind farm
SG 1
SG 2 SG 3
1
2 3
4
5 6
7 9
8
IEEE nine-bus system
wind farm
1
2
3
4
5
6
7
8
9
10
• high-fidelity models for turbines,
machines, & IEEE-9-bus system
• fast frequency response via
decentralized DeePC at turbines
h(g) = Proj
⇣
ud
yd
⌘
g
2
2
subspace ID + control
34/53
Slide 118
Slide 118 text
Case study ++ : wind farm
SG 1
SG 2 SG 3
1
2 3
4
5 6
7 9
8
IEEE nine-bus system
wind farm
1
2
3
4
5
6
7
8
9
10
• high-fidelity models for turbines,
machines, & IEEE-9-bus system
• fast frequency response via
decentralized DeePC at turbines
h(g) = Proj
⇣
ud
yd
⌘
g
2
2
subspace ID + control
34/53
Slide 119
Slide 119 text
DeePC is easy to implement ! try it !
! simple script adapted from our ETH Z¨
urich bachelor course on
Computational control : https://colab.research.google.com/
drive/1URdRqr-Up0A6uDMjlU6gwmsoAAPl1GId?usp=sharing
35/53
Slide 120
Slide 120 text
Towards a theory for nonlinear systems
36/53
Slide 121
Slide 121 text
Towards a theory for nonlinear systems
idea : lift nonlinear system to large/1-dimensional bi-/linear system
! Carleman, Volterra, Fliess, Koopman, Sturm-Liouville methods
! nonlinear dynamics can be approximated by LTI on finite horizon
36/53
Slide 122
Slide 122 text
Towards a theory for nonlinear systems
idea : lift nonlinear system to large/1-dimensional bi-/linear system
! Carleman, Volterra, Fliess, Koopman, Sturm-Liouville methods
! nonlinear dynamics can be approximated by LTI on finite horizon
regularization singles out relevant features / basis functions in data
36/53
Slide 123
Slide 123 text
Towards a theory for nonlinear systems
idea : lift nonlinear system to large/1-dimensional bi-/linear system
! Carleman, Volterra, Fliess, Koopman, Sturm-Liouville methods
! nonlinear dynamics can be approximated by LTI on finite horizon
regularization singles out relevant features / basis functions in data
https://www.research-collection.ethz.ch/handle/20.500.11850/493419
36/53
Slide 124
Slide 124 text
Works very well across case studies
37/53
Slide 125
Slide 125 text
regularization
m
robustification
Slide 126
Slide 126 text
Distributional robustification beyond LTI
• problem abstraction : minx2X c b
⇠, x
where b
⇠ denotes measured data
38/53
Slide 127
Slide 127 text
Distributional robustification beyond LTI
• problem abstraction : minx2X c b
⇠, x = minx2X E
⇠⇠b
P
⇥
c (⇠, x)
⇤
where b
⇠ denotes measured data with empirical distribution
b
P = b
⇠
38/53
Sen S See
E E E5
Slide 128
Slide 128 text
Distributional robustification beyond LTI
• problem abstraction : minx2X c b
⇠, x = minx2X E
⇠⇠b
P
⇥
c (⇠, x)
⇤
where b
⇠ denotes measured data with empirical distribution
b
P = b
⇠
) poor out-of-sample performance of above sample-average solution x?
for real problem: E⇠⇠P
⇥
c (⇠, x?)
⇤
where P is the unknown distribution of ⇠
38/53
Slide 129
Slide 129 text
Distributional robustification beyond LTI
• problem abstraction : minx2X c b
⇠, x = minx2X E
⇠⇠b
P
⇥
c (⇠, x)
⇤
where b
⇠ denotes measured data with empirical distribution
b
P = b
⇠
) poor out-of-sample performance of above sample-average solution x?
for real problem: E⇠⇠P
⇥
c (⇠, x?)
⇤
where P is the unknown distribution of ⇠
• distributionally robust formulation accounting for all (possibly nonlinear)
stochastic processes that could have generated the data
38/53
Slide 130
Slide 130 text
Distributional robustification beyond LTI
• problem abstraction : minx2X c b
⇠, x = minx2X E
⇠⇠b
P
⇥
c (⇠, x)
⇤
where b
⇠ denotes measured data with empirical distribution
b
P = b
⇠
) poor out-of-sample performance of above sample-average solution x?
for real problem: E⇠⇠P
⇥
c (⇠, x?)
⇤
where P is the unknown distribution of ⇠
• distributionally robust formulation accounting for all (possibly nonlinear)
stochastic processes that could have generated the data
inf
x2X
sup
Q2B✏(b
P)
E⇠⇠Q
⇥
c (⇠, x)
⇤
38/53
maximize over
all Q which are
"E-close"
to
my samples i
Slide 131
Slide 131 text
Distributional robustification beyond LTI
• problem abstraction : minx2X c b
⇠, x = minx2X E
⇠⇠b
P
⇥
c (⇠, x)
⇤
where b
⇠ denotes measured data with empirical distribution
b
P = b
⇠
) poor out-of-sample performance of above sample-average solution x?
for real problem: E⇠⇠P
⇥
c (⇠, x?)
⇤
where P is the unknown distribution of ⇠
• distributionally robust formulation accounting for all (possibly nonlinear)
stochastic processes that could have generated the data
inf
x2X
sup
Q2B✏(b
P)
E⇠⇠Q
⇥
c (⇠, x)
⇤
where B✏(b
P) is an ✏-Wasserstein ball
centered at empirical sample distribution b
P :
B✏(b
P) =
⇢
P : inf
⇧
Z
⇠ b
⇠
p
d⇧ ✏
ˆ
⇠
⇠
ˆ
P
P
⇧
38/53
Slide 132
Slide 132 text
• distributionally robustness ⌘ regularization : under minor conditions
Theorem: inf
x2X
sup
Q2B✏(b
P)
E⇠⇠Q
⇥
c (⇠, x)
⇤
| {z }
distributional robust formulation
39/53
Slide 133
Slide 133 text
• distributionally robustness ⌘ regularization : under minor conditions
Theorem: inf
x2X
sup
Q2B✏(b
P)
E⇠⇠Q
⇥
c (⇠, x)
⇤
| {z }
distributional robust formulation
⌘ min
x2X
c
⇣
b
⇠, x
⌘
+ ✏ Lip(c) · kxk?
p
| {z }
previous regularized DeePC formulation
39/53
Slide 134
Slide 134 text
• distributionally robustness ⌘ regularization : under minor conditions
Theorem: inf
x2X
sup
Q2B✏(b
P)
E⇠⇠Q
⇥
c (⇠, x)
⇤
| {z }
distributional robust formulation
⌘ min
x2X
c
⇣
b
⇠, x
⌘
+ ✏ Lip(c) · kxk?
p
| {z }
previous regularized DeePC formulation
Cor : `1
-robustness in trajectory space
() `1
-regularization of DeePC
39/53
Slide 135
Slide 135 text
• distributionally robustness ⌘ regularization : under minor conditions
Theorem: inf
x2X
sup
Q2B✏(b
P)
E⇠⇠Q
⇥
c (⇠, x)
⇤
| {z }
distributional robust formulation
⌘ min
x2X
c
⇣
b
⇠, x
⌘
+ ✏ Lip(c) · kxk?
p
| {z }
previous regularized DeePC formulation
Cor : `1
-robustness in trajectory space
() `1
-regularization of DeePC
10-5 10-4 10-3 10-2 10-1 100
0
0.5
1
1.5
2
2.5
3
3.5
Cost
105
cost
✏
39/53
Slide 136
Slide 136 text
• distributionally robustness ⌘ regularization : under minor conditions
Theorem: inf
x2X
sup
Q2B✏(b
P)
E⇠⇠Q
⇥
c (⇠, x)
⇤
| {z }
distributional robust formulation
⌘ min
x2X
c
⇣
b
⇠, x
⌘
+ ✏ Lip(c) · kxk?
p
| {z }
previous regularized DeePC formulation
Cor : `1
-robustness in trajectory space
() `1
-regularization of DeePC
10-5 10-4 10-3 10-2 10-1 100
0
0.5
1
1.5
2
2.5
3
3.5
Cost
105
cost
✏
• measure concentration: average matrix
1
N
PN
i=1 Hi(yd) from i.i.d. experiments
=) ambiguity set B✏(b
P) includes true P
with high confidence if ✏ ⇠ 1/N1/ dim(⇠)
N = 1
N = 10
39/53
Slide 137
Slide 137 text
Further ingredients
• more structured uncertainty sets :
tractable reformulations (relaxations)
& performance guarantees
40/53
Slide 138
Slide 138 text
Further ingredients
• more structured uncertainty sets :
tractable reformulations (relaxations)
& performance guarantees
• distributionally robust probabilistic constraints
sup
Q2B✏(b
P)
CVaRQ
1 ↵
() averaging + regularization + tightening
CVaRP
1 ↵
(X)
P(X) 1 ↵
VaRP
1 ↵
(X)
40/53
Slide 139
Slide 139 text
Further ingredients
• more structured uncertainty sets :
tractable reformulations (relaxations)
& performance guarantees
• distributionally robust probabilistic constraints
sup
Q2B✏(b
P)
CVaRQ
1 ↵
() averaging + regularization + tightening
CVaRP
1 ↵
(X)
P(X) 1 ↵
VaRP
1 ↵
(X)
• replace (finite) moving horizon estimation via (uini
yini
) by recursive Kalman
filtering based on optimization solution g? as hidden state ...
40/53
Slide 140
Slide 140 text
white elephant
Slide 141
Slide 141 text
white elephant: how does DeePC
perform against SysID + control ?
Slide 142
Slide 142 text
white elephant: how does DeePC
perform against SysID + control ?
surprise: DeePC consistently
beats (certainty-equivalence)
identification & control of LTI
models across all real case studies !
Slide 143
Slide 143 text
white elephant: how does DeePC
perform against SysID + control ?
surprise: DeePC consistently
beats (certainty-equivalence)
identification & control of LTI
models across all real case studies !
why ?!?
Slide 144
Slide 144 text
Comparison: direct vs. indirect control
indirect ID-based data-driven control
minimize control cost u, y
subject to u, y satisfy parametric model
where model 2 argmin id cost ud, yd
subject to model 2 LTI(n, `) class
41/53
Slide 145
Slide 145 text
Comparison: direct vs. indirect control
indirect ID-based data-driven control
minimize control cost u, y
subject to u, y satisfy parametric model
where model 2 argmin id cost ud, yd
subject to model 2 LTI(n, `) class
)
ID
41/53
Slide 146
Slide 146 text
Comparison: direct vs. indirect control
indirect ID-based data-driven control
minimize control cost u, y
subject to u, y satisfy parametric model
where model 2 argmin id cost ud, yd
subject to model 2 LTI(n, `) class
)
ID
ID projects data on
the set of LTI models
• with parameters (n, `)
• removes noise & thus
lowers variance error
• su ers bias error if
plant is not LTI(n, `)
41/53
Slide 147
Slide 147 text
Comparison: direct vs. indirect control
indirect ID-based data-driven control
minimize control cost u, y
subject to u, y satisfy parametric model
where model 2 argmin id cost ud, yd
subject to model 2 LTI(n, `) class
)
ID
ID projects data on
the set of LTI models
• with parameters (n, `)
• removes noise & thus
lowers variance error
• su ers bias error if
plant is not LTI(n, `)
direct regularized data-driven control
minimize control cost u, y + · regularizer
subject to u, y consistent with ud, yd data
41/53
Ce Im H(i)
Slide 148
Slide 148 text
Comparison: direct vs. indirect control
indirect ID-based data-driven control
minimize control cost u, y
subject to u, y satisfy parametric model
where model 2 argmin id cost ud, yd
subject to model 2 LTI(n, `) class
)
ID
ID projects data on
the set of LTI models
• with parameters (n, `)
• removes noise & thus
lowers variance error
• su ers bias error if
plant is not LTI(n, `)
direct regularized data-driven control
minimize control cost u, y + · regularizer
subject to u, y consistent with ud, yd data
• regularization robustifies
! choosing makes it work
• no projection on LTI(n, `)
! no de-noising & no bias
41/53
Slide 149
Slide 149 text
Comparison: direct vs. indirect control
indirect ID-based data-driven control
minimize control cost u, y
subject to u, y satisfy parametric model
where model 2 argmin id cost ud, yd
subject to model 2 LTI(n, `) class
)
ID
ID projects data on
the set of LTI models
• with parameters (n, `)
• removes noise & thus
lowers variance error
• su ers bias error if
plant is not LTI(n, `)
direct regularized data-driven control
minimize control cost u, y + · regularizer
subject to u, y consistent with ud, yd data
• regularization robustifies
! choosing makes it work
• no projection on LTI(n, `)
! no de-noising & no bias
hypothesis: ID wins in stochastic (variance) & DeePC in nonlinear (bias) case
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Slide 150
Slide 150 text
Case study: direct vs. indirect control
stochastic LTI case
• LQR control of 5th order LTI system
• Gaussian noise with varying noise to
signal ratio (100 rollouts each case)
• `1
-regularized DeePC, SysID via
N4SID, & judicious hyper-parameters
42/53
Slide 151
Slide 151 text
Case study: direct vs. indirect control
stochastic LTI case
• LQR control of 5th order LTI system
• Gaussian noise with varying noise to
signal ratio (100 rollouts each case)
• `1
-regularized DeePC, SysID via
N4SID, & judicious hyper-parameters
deterministic noisy 42/53
Slide 152
Slide 152 text
Case study: direct vs. indirect control
stochastic LTI case ! indirect ID wins
• LQR control of 5th order LTI system
• Gaussian noise with varying noise to
signal ratio (100 rollouts each case)
• `1
-regularized DeePC, SysID via
N4SID, & judicious hyper-parameters
deterministic noisy 42/53
Slide 153
Slide 153 text
Case study: direct vs. indirect control
stochastic LTI case ! indirect ID wins
• LQR control of 5th order LTI system
• Gaussian noise with varying noise to
signal ratio (100 rollouts each case)
• `1
-regularized DeePC, SysID via
N4SID, & judicious hyper-parameters
deterministic noisy
nonlinear case
• Lotka-Volterra + control: x+ = f(x, u)
• interpolated system
x+ = ✏·flinearized(x, u)+(1 ✏)·f(x, u)
• same ID & DeePC as on the left
& 100 initial x0
rollouts for each ✏
42/53
Slide 154
Slide 154 text
Case study: direct vs. indirect control
stochastic LTI case ! indirect ID wins
• LQR control of 5th order LTI system
• Gaussian noise with varying noise to
signal ratio (100 rollouts each case)
• `1
-regularized DeePC, SysID via
N4SID, & judicious hyper-parameters
deterministic noisy
nonlinear case
• Lotka-Volterra + control: x+ = f(x, u)
• interpolated system
x+ = ✏·flinearized(x, u)+(1 ✏)·f(x, u)
• same ID & DeePC as on the left
& 100 initial x0
rollouts for each ✏
nonlinear linear
42/53
Slide 155
Slide 155 text
Case study: direct vs. indirect control
stochastic LTI case ! indirect ID wins
• LQR control of 5th order LTI system
• Gaussian noise with varying noise to
signal ratio (100 rollouts each case)
• `1
-regularized DeePC, SysID via
N4SID, & judicious hyper-parameters
deterministic noisy
nonlinear case ! direct DeePC wins
• Lotka-Volterra + control: x+ = f(x, u)
• interpolated system
x+ = ✏·flinearized(x, u)+(1 ✏)·f(x, u)
• same ID & DeePC as on the left
& 100 initial x0
rollouts for each ✏
nonlinear linear
42/53
Slide 156
Slide 156 text
Power system case study revisited
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time (s)
uncontrolled flow (p.u.)
• complex 4-area power system: large (n = 208), few measurements (8),
nonlinear, noisy, sti , input constraints, & decentralized control
• control objective: damping of inter-area oscillations via HVDC link
• real-time MPC & DeePC prohibitive ! choose T, Tini
, & Tfuture
wisely
43/53
Slide 157
Slide 157 text
Centralized control
0 5 10 15 20 25 30
0.2
0.4
0.6
0.8
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
time (s)
a control; —– with PEM-MPC (s = 60); —– with DeePC
Closed‐loop cost
Number of simulations
DeePC
PEM‐MPC
ost comparison of DeePC and PEM-MPC under the practic
= Prediction Error
Method (PEM)
System ID + MPC
t < 10 s : open loop
data collection with
white noise excitat.
t > 10 s : control
44/53
Slide 158
Slide 158 text
Performance: DeePC wins (clearly!)
Closed‐loop cost
Number of simulations
DeePC
PEM‐MPC
Measured closed-loop cost =
P
k
kyk
rk
k2
Q
+ kuk
k2
R
45/53
Slide 159
Slide 159 text
DeePC hyper-parameter tuning
Closed‐loop cost
Closed‐loop cost
Tfuture
regularizer
g
• for distributional robustness
⇡ radius of Wasserstein ball
• wide range of sweet spots
! choose g = 20
estimation horizon Tini
• for model complexity ⇡ lag
• Tini 50 is su cient & low
computational complexity
! choose Tini = 60
46/53
-4
I I
Slide 160
Slide 160 text
Closed‐loop cost
Closed‐loop cost
Closed‐loop cost
Closed‐loop cost
Tfuture
prediction horizon Tfuture
• nominal MPC is stable if
horizon Tfuture
long enough
! choose Tfuture = 120 &
apply first 60 input steps
data length T
• long enough for low-rank
condition but card(g) grows
! choose T = 1500
(data matrix ⇡ square)
47/53
Slide 161
Slide 161 text
Computational cost
time (s)
0 5 10 15 20 25 30
0.2
0.4
0.6
0.8
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
• T = 1500
• g = 20
• Tini = 60
• Tfuture = 120 & apply
first 60 input steps
• sampling time = 0.02 s
• solver (OSQP) time = 1 s
(on Intel Core i5 7200U)
) implementable
48/53
Slide 162
Slide 162 text
Comparison: Hankel & Page matrix
Control Horizon k Control Horizon k
Averaged Closed‐loop Cost
S0=1
⌅ Hankel matrix
⌅ Hankel matrix with
SVD ( threshhold = 1)
⌅ Page matrix
⌅ Page matrix with
SVD ( threshhold = 1)
• comparison baseline: Hankel and Page matrices of same size
• perfomance : Page consistency beats Hankel matrix predictors
• o ine denoising via SVD threshholding works wonderfully for
Page though obviously not for Hankel (entries are constrained)
• e ects very pronounced for longer horizon (= open-loop time)
• price-to-be-paid : Page matrix predictor requires more data
49/53
Slide 163
Slide 163 text
Decentralized implementation
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time (s)
uncontrolled flow (p.u.)
• plug’n’play MPC: treat interconnection P3
as disturbance variable w
with past disturbance wini
measurable & future wfuture 2 W uncertain
• for each controller augment trajectory matrix with disturbance data w
• decentralized robust min-max DeePC: ming,u,y maxw2W
50/53
min mu
Slide 164
Slide 164 text
Decentralized control performance
0 5 10 15 20 25 30
0.2
0.4
0.6
0.8
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
time (s)
• colors correspond
to di erent hyper-
parameter settings
(not discernible)
• ambiguity set W
is 1-ball (box)
• for computational
e ciency W is
downsampled
(piece-wise linear)
• solver time ⇡ 2.6 s
) implementable
51/53
Slide 165
Slide 165 text
Conclusions
main take-aways
• matrix time series as predictive model
• robustness & side-info by regularization
• method that works in theory & practice
• focus is robust prediction not predictor ID
SG 1
SG 2 SG 3
1
2 3
4
5 6
7 9
8
IEEE nine-bus system
wind farm
1
2
3
4
5
6
7
8
9
10
52/53
Slide 166
Slide 166 text
Conclusions
main take-aways
• matrix time series as predictive model
• robustness & side-info by regularization
• method that works in theory & practice
• focus is robust prediction not predictor ID
ongoing work
! certificates for adaptive & nonlinear cases
! applications with a true “business case”,
push TRL scale, & industry collaborations SG 1
SG 2 SG 3
1
2 3
4
5 6
7 9
8
IEEE nine-bus system
wind farm
1
2
3
4
5
6
7
8
9
10
52/53
Slide 167
Slide 167 text
Conclusions
main take-aways
• matrix time series as predictive model
• robustness & side-info by regularization
• method that works in theory & practice
• focus is robust prediction not predictor ID
ongoing work
! certificates for adaptive & nonlinear cases
! applications with a true “business case”,
push TRL scale, & industry collaborations SG 1
SG 2 SG 3
1
2 3
4
5 6
7 9
8
IEEE nine-bus system
wind farm
1
2
3
4
5
6
7
8
9
10
questions we should discuss
• catch? violate no-free-lunch theorem ? ! more real-time computation
52/53
- LQR
Slide 168
Slide 168 text
Conclusions
main take-aways
• matrix time series as predictive model
• robustness & side-info by regularization
• method that works in theory & practice
• focus is robust prediction not predictor ID
ongoing work
! certificates for adaptive & nonlinear cases
! applications with a true “business case”,
push TRL scale, & industry collaborations SG 1
SG 2 SG 3
1
2 3
4
5 6
7 9
8
IEEE nine-bus system
wind farm
1
2
3
4
5
6
7
8
9
10
questions we should discuss
• catch? violate no-free-lunch theorem ? ! more real-time computation
• when does direct beat indirect ? ! Id4Control & bias/variance issues ?
52/53
QR
&
of objective should bias ID
Slide 169
Slide 169 text
53/53
Slide 170
Slide 170 text
Florian’s version of
53/53
Slide 171
Slide 171 text
Thanks !
Florian D¨
orfler
mail: dorfl[email protected]
[link] to homepage
[link] to related publications